Stanley's Lemma and Multiple Theta Functions

William Y. C. Chen and Lisa H. Sun

  Abstract:   We present an algorithmic approach to the verification of identities on multiple theta functions in the form of products of theta functions [(-1)δa1α1a2α2... arαrqs qt], where αi are integers, δ=0 or 1, s∈ Q, t∈ Q+, and the exponent vectors (α1,, α2,...,αr) are linearly independent over Q. For an identity on such multiple theta functions, we provide an algorithmic approach for computing a system of contiguous relations satisfied by all the involved multiple theta functions. Using Stanley's Lemma on the fundamental parallelepiped, we show that a multiple theta function can be determined by a finite number of its coefficients. Thus such an identity can be reduced to a finite number of simpler relations. Many classical multiple theta function identities fall into this framework, including Riemann's addition formula and the extended Riemann identity.

  AMS Classification:  05E45, 14K25

  theta function, multiple theta function, contiguous relation, Jacobi's triple prod- uct identity, addition formula, Stanley's Lemma

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